Question: Simplify the following expression: $n = \dfrac{6r^2 + 12r}{-30r^2 - 30r}$ You can assume $r \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $6r^2 + 12r = (2\cdot3 \cdot r \cdot r) + (2\cdot2\cdot3 \cdot r)$ The denominator can be factored: $-30r^2 - 30r = - (2\cdot3\cdot5 \cdot r \cdot r) - (2\cdot3\cdot5 \cdot r)$ The greatest common factor of all the terms is $6r$ Factoring out $6r$ gives us: $n = \dfrac{(6r)(r + 2)}{(6r)(-5r - 5)}$ Dividing both the numerator and denominator by $6r$ gives: $n = \dfrac{r + 2}{-5r - 5}$